Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

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1answer
34 views

Showing $\left(\sum\limits_{j=0}^n a_jx^j\right)\left(\sum\limits_{k=0}^n b_kx^k\right)=…$ for certain properties

Show that $\left(\sum\limits_{j=0}^n a_jx^j\right)\left(\sum\limits_{k=0}^n b_kx^k\right)=\sum\limits_{m=0}^{2n}\left(\sum\limits_{j+k=m} a_jb_k\right)x^m=\sum\limits_{m=0}^{2n}(a_0b_m+a_1b_{m-1}+...+...
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1answer
32 views

How to prove this series Cauchy product?

Let the first series be $\sum_{k=0}^{\infty} u_{k}(x-a)^{k}$. Let the second series be $\sum_{k=0}^{\infty} v_{k}(x-a)^{k}$ Their convergence areas are $R_{1}$ and $R_{2}$. I don't know how to ...
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2answers
79 views

Cauchy product summation converges

I had a previous question here, which I'm quoting: How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1) (n-k+1)}$$ I tried to prove ...
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1answer
32 views

Cauchy Product of two geometric series.

So I wanted to calculate/prove that $$\frac{1}{(1-q)^2} = \displaystyle\sum_{n=0}^{\infty}(n+1)q^n,$$ $q\in \mathbb{C},$ $\mid q \mid < 1$ using the Cauchy-Product. So this is my solution and I'm ...
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1answer
31 views

show the power series converges for |x|<1, and find the closed forms

\begin{matrix} 1) S(x) = \sum_{n=0}^\infty (1+ \frac{1}{1!} + \frac{1}{2!} + ...+ \frac{1}{n!}) x^n \\ 2) S(x) =\sum_{n=1}^\infty a_n x^n, a_n= \Big\{ \begin{alignedat}{3} -2/n ,n=3k \\ 1/n,n \neq 3k ...
2
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1answer
72 views

Implementation of Cauchy product on $\cos x\cdot \sin x$

I have a mistake that I can't find somewhere along the way. Please help me find the place things go wrong. Find the product of $\cos x$ and $\sin x$ as defined: $$\cos(x) = \sum_{k=0}^{\infty} \frac{\...
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0answers
26 views

Cauchy product of 2 series

Let $z \in \mathbb{C}$. I want to prove that $$\sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} \sum_{n\in\mathbb{Z}} \frac{(-1)^n}{z-n} = \sum_{n\in\mathbb{Z}} \frac{1}{(z-n)^2}.$$ Using the Cauchy Product ...
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1answer
30 views

How to show that these sums are equal?

Assuming that the product is associative, I would like to show that $$ \sum_{s=0}^{k} (\sum_{i=0}^{s} (\alpha_i \cdot \beta_{s-i}) \cdot \gamma_{k-s}) = \sum_{s=0}^{k} ( \sum_{i=0}^{k-s} \alpha_s \...
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3answers
44 views

Product of two generating functions in terms of the ordinary generating function? (recurrence relation solving)

I have the following recurrence relation that I am trying to solve: $a_0 = 1$ $$ a_n = \sum_{i = 0}^{n-1} (i+1)a_i$$ for $ n \geq 1$. Let's define $A(x) = \sum_{n= 0}^\infty a_nx^n$, so multiplying ...
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1answer
16 views

Cauchy product for power series of different powers

How would you find the Cauchy product of two power series of different powers? For example, I want to find the Cauchy product of the two series $ \exp(x) = \sum_{k=0}^{\infty}\frac{x^k}{k!}$ and $ \...
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2answers
63 views

An example: a convergence series, a divergent series, whose Cauchy product is convergent.

How to find an example: a convergence series $\sum a_n$, a divergent series $\sum b_n$, whose Cauchy product $\sum c_n$ with $c_n=\sum_{i+j=n}a_ib_j$ is convergent? Is there a simple example?
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1answer
37 views

The Cauchy Product : Calculate the coefficients, $c_n$, in the product.

I found this definition to be different from other ones and the answer makes me feel weird. If $$A_{N}(x)=\sum_{n=0}^{n=N}a_{n}x^n$$ and $$B_{N}(x)=\sum_{n=0}^{n=N}b_{n}x^n,$$ calculate the ...
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0answers
140 views

Are the unconditionally convergent series, with terms in a Banach algebra, closed under the Cauchy product?

We have a Banach algebra $\mathbb L$, and two sequences $(A_0,A_1,A_2,\cdots),\;(B_0,B_1,B_2,\cdots)\in\mathbb L^{\mathbb N}$, for which the sums $\sum_{n\in\mathbb N}A_n,\;\sum_{n\in\mathbb N}B_n$ ...
7
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1answer
128 views

The necessity of absolute convergence in the convergence of the Cauchy product of series?

The Mertens' theorem claims that Suppose $\sum_{n=0}^\infty a_n,\sum_{n=0}^\infty b_n$ are two convergent series of complex numbers, convergent to $A,\beta$ respectively. If $\sum_na_n$ converges ...
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1answer
67 views

Is the sequence space $\ell^p$ closed under the Cauchy product?

Given that two formal power series have coefficients in $\ell^p$, $$a(x)=\sum_{n=0}^\infty a_nx^n,\quad\sum_{n=0}^\infty|a_n|^p<\infty$$ $$b(x)=\sum_{n=0}^\infty b_nx^n,\quad\sum_{n=0}^\infty|b_n|...
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3answers
71 views

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$

Is there a way to re-write $\sum_{n=1}^{\infty}{\left(\sum_{k=1}^{n}{\frac{1}{k}}\right)}z^n$ I was thinking that I could use the cauchy product, but therefore I have to achieve this:$$\left(\sum_{...
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2answers
35 views

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$

How to show that $\sum_{k=0}^{\infty}{(k+1)z^k}=\frac{1}{(1-z)^2}$ for $z\in \mathbb{C}:|z|<1$ I believe that I have to use the cauchy product? But how do transform the expression to be a product ...
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1answer
49 views

Product of Two Summations with Different Upper Limits

I'm trying to multiply two different finite summations with different upper limits. I've tried Cauchy Product but i think it's valid for same upper limits. I also tried to split the summation. Any ...
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3answers
53 views

Find the power series of $f(x)=\frac{x}{x+1}$ in $x_0=0$

In 1st Semester Calculus book I found an exercise that asks me to find the above power series of the function at the point $x_0 = 0$ using the geometric series formula and the Cauchy-Product. So far I'...
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1answer
88 views

Cauchy product and geometric series [duplicate]

I was given this series: Let $q \in \mathbb{C}, \mid q\mid <1. $ $$\frac{1}{2}\sum_{n=0}^{\infty} (n^2 +3n +2)q^n $$ Now I have to show that $$\frac{1}{(1-q)^3}=\frac{1}{2}\sum_{n=0}^{\infty} (...
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1answer
198 views

Multiplying two polynomials: explanation of the general formula for the coefficients

If $$f(x)=a_nx^n+...+a_0$$ and $$g(x)=b_mx^m+…+b_0$$ then $$f(x)\cdot g(x)=c_{m+n}x^{m+n}+...+c_0$$ where $c_k=\sum_{r+s=k}a_rb_s,\quad k=0,....,m+n$ I know that the degree $0$ and $(m+n)$ exists, ...
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2answers
81 views

Generating Function For Catalan Numbers Type Sequence

I've been working my way through an old post, but I don't think the solution offered can be correct. The question is; Find the generating function (within a choice of sign) for: $$c_{n+1} = 2\sum_{k=...
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0answers
47 views

Cauchy product of power series with $x^{2n}$

I am trying to rewrite a function $y(x)=\frac{1}{1+x+x^2+x^3}$ as a series. I used geometric series and got to two power series that I need to Cauchy product, however, one of them has $x^{2n}$ and I ...
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0answers
135 views

Infinitely Nested Infinite Series / Infinite Composition of Series

Is there any documentation about a series like this? What is it called? Does it have a value? I have tried several searches and couldn't find anything close to this. Any guidance would be ...
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0answers
44 views

Cauchy product for the reciprocal of the polynomial $x - 2x^2 + 3x^3 - 4x^4$

I have come across some Laurent series in which the denominator of a fraction contains a power series. Looking around I came across this Calculate Laurent series for $1/ \sin(z)$ which suggests that ...
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1answer
32 views

How was Merten's theorem proven on the step comparing the convergence of one sequence and the Cauchy product partial sum?

I'm trying to understand step 3 of the proof of Merten's theorem. How is it known that $N*_{sup}|B_i-B|$ term converges slow enough that there is an N where equation 3 is true: $|a_n|\le\frac{\...
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2answers
156 views

Proving $\sin(2x) = 2\sin(x)\cos(x)$ using Cauchy product

I've stumbled upon this question. I can't understand why all even terms of the Cauchy product are $0$, since we add only positive numbers: \begin{equation*} c_n = \begin{cases} \sum\limits_{k=0}^{m} \...
5
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3answers
198 views

On the series expansion of $\frac{\operatorname{Li}_3(-x)}{1+x}$ and its usage

Recently I have asked about the evaluation of an integral involving a Trilogarithm $($which can be found here$)$. Pisco provided a quite elegant approach starting with a functional equation of the ...
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2answers
88 views

Simplifying $\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$

I have come across a sum of the following form; $$\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{l=0}^{\min(n,m)}a_{l}b_{m-l}c_{n-l}$$ and want to simplify it (in particular to remove the $min(n,m)$). ...
6
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0answers
376 views

Series whose Cauchy product is absolutely convergent - A general example

Is there series that is divergent or conditionally convergent with absolutely convergent Cauchy product? Seems like there is a group of these examples! Perhaps finding divergent series with ...
2
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1answer
220 views

Conditionally converges $\sum_{k=0}^\infty a_n$, $\sum_{k=0}^\infty b_n$ and also their Cauchy product

Edited: I post a new post here that is somewhat related to this question. It proved that: If $\sum_{k=0}^\infty a_n$ and $\sum_{k=0}^\infty b_n$ converges conditionally (not absolutely), then their ...
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1answer
39 views

Calculating the limit based on the cauchy product

I'm attempting to find the limit of $$ \sum^\infty_{k=0} k^2 q^k$$ using the result of the cauchy product $$ \sum^\infty_{k=0} k q^k * \sum^\infty_{k=0} q^k$$ and I have calculated the cauchy ...
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3answers
113 views

Summation Recurrence Relation

How to solve this Summation Recurrence Relation: $$x_n=\sum_{i=1}^n a_ix_{n-i}\,,\,\,\,n\ge1$$where, $x_0=1$ and $a_n$ is some arbitrary sequence. The right hand side of the recurrence looks ...
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1answer
118 views

Radius of convergence of Cauchy product – examples

Assume two power series $\sum_{n\ge0}a_n x^n=f_a(x),\sum_{n\ge0}b_n x^n=f_b(x)$ with radii $r_a,r_b$ (respectively) and $r_a\lneqq r_b.$ Consider their Cauchy product $$\sum_{n\ge0}\left(\sum_{k=0}^n ...
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2answers
125 views

Find the Cauchy product of $e^x$ and $e^{-x}$.

I was able to get to: $c_n$ = $$\sum_{k=0}^{n} \binom{n}{k} \frac{x^k (-x)^{n-k}}{n!}$$ but I am stuck as to how to get past this. Is there a property of the series that lets me pull the negative ...
5
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2answers
173 views

How to get sums like these in the form of the Cauchy Product?

I know that the question isn't very well-worded, please feel free to change it to something better. I have this sum: $$\sum_{k=0}^{\infty} \sum_{l=0}^{k} a_lx^ll!a_{k-l}x^{k-l+1}(k-l)!\frac{1}{(k+1)!...
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1answer
228 views

Counterexample to Cauchy product theorem

The Cauchy product theorem for infinite series of complex numbers states if $\sum a_n$ and $\sum b_n$ are two absolutely convergent series then the Cauchy product $\sum c_n$, where $c_n=\sum_{p+q=n} ...
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0answers
72 views

Cauchy Product of Two Divergent Series

Problem Examine the following Cauchy product and the factors for convergence and in the case of convergence determine the limit. $$\left(3 + \sum^{\infty}_{k=1}3^k\right)\left(-2+\sum^{\infty}_{k=1}...
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0answers
20 views

Unclear equation in my scriptum (perhaps cauchy-product) [duplicate]

Hi the following equation in my scriptum seems unclear. I think it has something to do with cauchy-product but i dont know
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1answer
109 views

How to recast these two exponential of infinite power series as simple power series?

I have two exponential of infinite power series, with different expressions for coefficients $a_n$, that I would like to recast as two other power series without the exponential $$\exp\left(\sum_{n=1}^...
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4answers
57 views

Power series of $\frac{1}{(1-z)^2}$

I want to show, that the following is true for every $z\in C$ with $|z|<1$: $$\frac{1}{(1-z)^2} =\sum_{k=1}^\infty kz^{k-1}$$ I think there is a way with the Cauchy-Product
1
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1answer
57 views

Let $G(x)=\frac{1}{(1-x)^2}$. Prove that $G(x)=\sum_{n=0}^{\infty}(n+1)x^n$.

Let $G(x)=\frac{1}{(1-x)^2}$. Prove that $$G(x)=\sum_{n=0}^{\infty}(n+1)x^n.$$ The solution given uses the Cauchy Product. It is shown below: $$\begin{aligned} G(x)&=\left(\sum_{n=0}^{\infty}x^n\...
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0answers
712 views

Power Series Solution for an ODE which has trigonometric coefficient functions

The ODE for which we seek a power series solution is: $$y''+ \cos(x)y' + x\sin(x)y = 0,\hspace{0.4cm} y(0) = 1,\hspace{.1cm} y'(0) = 0$$ I need to find the partial sum up to five, from the initial ...
4
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3answers
381 views

Cauchy Product starting from $1$

The definition of the Cauchy product from Wikipedia is defined as $$\left(\sum_{i=0}^\infty a_i\right)\left(\sum_{j=0}^\infty b_j\right) =\sum_{k=0}^\infty\sum_{\ell=0}^ka_\ell b_{k-\ell}$$ ...
3
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2answers
67 views

show that $\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\right) = \sum_{k=0}^\infty\frac{(a+b)^k}{k!}$

I need to show that if $\sum_{i=0}^\infty \frac{a^i}{i!}$ is absolutely convergent for all $a\in\mathbb{R}$, then $$\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\...
1
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0answers
98 views

Decomposition in product

I start my question with an example that I did. Give the following series $R(x):=\sum_{d=1}^{\infty}(\sum_{l=0}^{d-1} (-1)^{l}\frac{1}{ \ \ell!(d-1-\ell)!})q_{1}^{d} x^d$. I could decompose this as a ...
3
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2answers
154 views

Calculate sum of a series

How to find the sum of $$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\left\{\sum_{k=1}^{2n+1}\frac{(-1)^k} k \right\}$$ $$\begin{array}\\ \frac{1}{1}&\times&(-\frac{1}{1}) &+&\ \ \\ (-\...
5
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4answers
185 views

Find limit of sum

I suspect that $\lim_{n \to \infty} \sum_{k = 0}^{n - 1}\frac{k}{a^k(n - k)} = 0$ for $a > 1$. I know that this product represents the taylor coefficients of $\frac{-ax\ln(1 - x)}{(a - x)^2}$ by ...
3
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3answers
116 views

The series $\sum_{k=1}^\infty\sum_{n=1}^k k^{-3}/(2n-1)$

$$\sum_{k=1}^\infty\sum_{n=1}^k\frac{1}{(2n-1)k^3}$$ Can anyone help me find this series? I tried to use Cauchy product but I don't know how I can complete it.
5
votes
2answers
666 views

Convergent Cauchy product of divergent series

I was looking at the counterexample $a_n = b_n = (-1)^n/\sqrt{n}$ where $\sum a_n$ and $\sum b_n$ converge but the Cauchy product $\sum_{k=0}^\infty c_k = \sum_{k=0}^\infty \sum_{j=0}^{k}a_j b_{k-j}$ ...